Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

Table of contents

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[311.] THEOR. XLII. PROP. LXVIII.
Page: 274 (88)
[312.] COROLL. I.
Page: 276 (90)
[313.] COROLL. II.
Page: 276 (90)
[314.] MONITVM.
Page: 276 (90)
[315.] DEFINITIONES. I.
Page: 278 (92)
[316.] II.
Page: 278 (92)
[317.] III.
Page: 279 (93)
[318.] IIII.
Page: 279 (93)
[319.] PROBL. XIV. PROP. LXIX.
Page: 280 (94)
[320.] SCHOLIVM I.
Page: 282 (96)
[321.] COROLL. I.
Page: 282 (96)
[322.] SCHOLIVM II.
Page: 282 (96)
[323.] COROLL. II.
Page: 282 (96)
[324.] SCHOLIVM III.
Page: 283 (97)
[325.] COROLL. III.
Page: 283 (97)
[326.] THEOR. XLIII. PROP. LXX.
Page: 284 (98)
[327.] COROLL.
Page: 286 (100)
[328.] THEOR. XLIV. PROP. LXXI.
Page: 286 (100)
[329.] COROLL.
Page: 287 (101)
[330.] THEOR. XLV. PROP. LXXII.
Page: 288 (102)
[331.] SCHOLIVM.
Page: 288 (102)
[332.] THEOR. XLVI. PROP. LXXIII.
Page: 288 (102)
[333.] THEOR. XLVII. PROP. LXXIV.
Page: 288 (102)
[334.] MONITVM.
Page: 290 (104)
[335.] LEMMA XIV. PROP. LXXV.
Page: 290 (104)
[336.] SCHOLIVM.
Page: 291 (105)
[337.] LEMMA XV. PROP. LXXVI.
Page: 292 (106)
[338.] THEOR. XLVIII. PROP. LXXVII.
Page: 292 (106)
[339.] MONITVM.
Page: 293 (107)
[340.] THEOR. IL. PROP. LXXVIII.
Page: 294 (108)
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            ſuntque in triangulis D F A, D F L anguli ad F æquales, cum ſint recti,
              <lb/>
            & </s>
            <s xml:id="echoid-s8742" xml:space="preserve">latus F D commune, atque angulus A D F minor eſt angulo L D F,
              <lb/>
            quare & </s>
            <s xml:id="echoid-s8743" xml:space="preserve">latus A F minus erit latere F L, & </s>
            <s xml:id="echoid-s8744" xml:space="preserve">A H eò minus F L; </s>
            <s xml:id="echoid-s8745" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-0315-01" xlink:href="note-0315-01a" xml:space="preserve">92. h.</note>
            bit ergo H F ad F L minorem rationem, quàm eadem F H ad H A, & </s>
            <s xml:id="echoid-s8746" xml:space="preserve">
              <lb/>
            componendo H L ad L F, ſiue G H ad D F, minorem quàm F A ad A
              <lb/>
            H, vnde rectangulum G H A ſub extremis minus erit rectangulo D F
              <note symbol="b" position="right" xlink:label="note-0315-02" xlink:href="note-0315-02a" xml:space="preserve">16. ſept.
                <lb/>
              Pappi.</note>
            ſub medijs, & </s>
            <s xml:id="echoid-s8747" xml:space="preserve">hoc ſemper, vbicunque ſumptum ſit punctum G, vel in-
              <lb/>
            ter D, & </s>
            <s xml:id="echoid-s8748" xml:space="preserve">K, vel in ipſo K, nempe rectangulum ad G, vel K, pertin-
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            gens, minus eſſe rectangulo D F A, ſiue D F A maius eſſe quocunque
              <lb/>
            prædictorum rectangulorum G H A, vel K C A, &</s>
            <s xml:id="echoid-s8749" xml:space="preserve">c.</s>
            <s xml:id="echoid-s8750" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8751" xml:space="preserve">Si autem punctum ſumatur in quadrante A K, vt in O; </s>
            <s xml:id="echoid-s8752" xml:space="preserve">demiſſa per-
              <lb/>
            pendiculari O P. </s>
            <s xml:id="echoid-s8753" xml:space="preserve">Cum ſit K C maior O P, & </s>
            <s xml:id="echoid-s8754" xml:space="preserve">C A maior A P, erit re-
              <lb/>
            ctangulum K C A maius rectangulo O P C, ſed rectangulum D F A oſtẽ-
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            ſum eſt maius rectangulo K C A, ergo rectangulum D F A eò amplius
              <lb/>
            maius erit rectangulo O P A.</s>
            <s xml:id="echoid-s8755" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s8756" xml:space="preserve">Si denique punctum ſumatur in peripheriæ ſextante D B, veluti in Q,
              <lb/>
            demiſſa perpendiculari Q R, & </s>
            <s xml:id="echoid-s8757" xml:space="preserve">iuncta D Q, & </s>
            <s xml:id="echoid-s8758" xml:space="preserve">producta, ipſa conueniet
              <lb/>
            o mnino cum diametro A B ad partes B, vt in S, quoniam angulus E D
              <lb/>
            Q eſt in portione E A Q ſemi - circulo maiori, ac propterea acutus, & </s>
            <s xml:id="echoid-s8759" xml:space="preserve">
              <lb/>
            angulus D F S rectus eſt, &</s>
            <s xml:id="echoid-s8760" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8761" xml:space="preserve">Et cum arcus A I E æqualis ſit arcui D B
              <lb/>
            E, vterque enim eſt triens peripheriæ, erit arcus A I E maior arcu Q B
              <lb/>
            E, ac ideo angulus A D E, vel A D F maior angulo Q D E, vel S D F,
              <lb/>
            ſed in triangulis A F D, S F D latus F D eſt commune, & </s>
            <s xml:id="echoid-s8762" xml:space="preserve">anguli ad F
              <lb/>
            ſunt æquales, eò quod ſint recti, & </s>
            <s xml:id="echoid-s8763" xml:space="preserve">angulus A D F maior eſt angulo S
              <lb/>
            D F, vnde latus A F maius eſt latere F S, & </s>
            <s xml:id="echoid-s8764" xml:space="preserve">adhuc maius latere R
              <note symbol="c" position="right" xlink:label="note-0315-03" xlink:href="note-0315-03a" xml:space="preserve">92. h.</note>
            habebit ergo F R ad R S maiorem rationem quàm eadem R F ad F A, & </s>
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            componendo F S ad S R, vel D F ad Q R, maiorem quàm R A ad A F;
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            <s xml:id="echoid-s8766" xml:space="preserve">vnde rectangulum D F A ſub extremis, maius erit rectangulo Q R
              <note symbol="d" position="right" xlink:label="note-0315-04" xlink:href="note-0315-04a" xml:space="preserve">16. ſept.
                <lb/>
              Pappi.</note>
            ſub medijs, & </s>
            <s xml:id="echoid-s8767" xml:space="preserve">hoc ſemper vbicunque aſſumptum ſit punctum Q in ſex-
              <lb/>
            tante D B. </s>
            <s xml:id="echoid-s8768" xml:space="preserve">Quare cum rectangulum A F D demonſtratum ſit maius om-
              <lb/>
            nium applicatosum, tum in triente A D, tum in ſextante D B, ipſum A
              <lb/>
            F D erit _MAXIMV M_, & </s>
            <s xml:id="echoid-s8769" xml:space="preserve">ſumptis duplis, rectangulum ſub ſagitta A F in
              <lb/>
            chordam D E, erit _MAXIMV M_ rectangulum ſub qualibet alia ſagitta in
              <lb/>
            ſuam chordam. </s>
            <s xml:id="echoid-s8770" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s8771" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8772" xml:space="preserve">Quodque alibi aliter enodabimus.</s>
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            <s xml:id="echoid-s8774" xml:space="preserve">2. </s>
            <s xml:id="echoid-s8775" xml:space="preserve">AD pleniorem autem doctrinã, in proxima ſequenti ſecunda figura, ma-
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            nentibus poſitione ijſdem punctis K, D, E, dico talium rectang lo-
              <lb/>
            rum id, quod puncto D propinquius eſt, ſemper maius eſſe remotiori.</s>
            <s xml:id="echoid-s8776" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8777" xml:space="preserve">Nam de ijs, quæ ad arcum quadrantis A K pertingunt, vtputa de re-
              <lb/>
            ctangulis A C K, A F R, A H G, &</s>
            <s xml:id="echoid-s8778" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8779" xml:space="preserve">patet A C K propinquius puncto D
              <lb/>
            maius eſſe rectangulo A F R, quod ab ipſo D magis remouetur, & </s>
            <s xml:id="echoid-s8780" xml:space="preserve">A F
              <lb/>
            R maius eſſe A H G, &</s>
            <s xml:id="echoid-s8781" xml:space="preserve">c. </s>
            <s xml:id="echoid-s8782" xml:space="preserve">cum, tum altitudines K C, R F, G H, tum ba-
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            ſes C A, F A, H A continuè decreſcant.</s>
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            <s xml:id="echoid-s8784" xml:space="preserve">De ijs verò, quæ perueniunt ad arcum K D, videlicet in punctis I,
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            L, ita ratiocinabimur. </s>
            <s xml:id="echoid-s8785" xml:space="preserve">Demittantur ex I, L ad diametrum perpendicu-
              <lb/>
            lares I M N, L O P, & </s>
            <s xml:id="echoid-s8786" xml:space="preserve">iungatur I L, quæ producta conueniet ad partes
              <lb/>
            L cum diametro in Q (nam arcus N A L maior eſt ſemi-peripheria, </s>
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